I am a chess novice.
Can someone suggest a good, free chess computer game that I can use to start to learn chess by playing against the computer? Maybe with some different difficult levels...
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It depends on which operating system you are running. If you are running a Linux OS, there are many, many options for you to choose from. One of the more-popular options is glChess.
It has a few features that make it a good game to learn on:
While it might not look as pretty as other games, it has plenty of options:
While very feature-rich, XBoard is probably a good one to get once you have a few games under your belt.
If you are on a Windows OS (Windows 7 or Vista) Chess Titans is good for starting-out, too. While probably scoffed at by many on this forum, the advantages of Chess Titans are:
You could check out http://lichess.org as a free online resource. They have general chess discussion forums and the ability to observe previous or current games. Also you can play against a computer (at multiple difficulties) and other players.
I would suggest going to chess.com and creating a free account there. You can play against computer with different difficulty levels, as well as against other players. At the same time there is loads of learning materials readily available for you (Book Openings, Tactical Trainer, Online Chess Mentor), which should prove very useful. You can also buy a premium account for 7$ per month, which will unlock even more learning opportunities like videos and etc, but a free account should be enough for the beginning.
You can play against computer for free online at http://www.apronus.com/chess/computer.htm - you can modify the strength of the computer by giving it less milliseconds to think per move. You can also set up different positions there and play them against compputer, like for example you can have the computer play as black without a queen.